File size: 3,747 Bytes
e5a97c6 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 | import Mathlib
import OAI.Combinatorics.MatchingCount.Complexity.OnlineInitialCost
namespace OAI
namespace MatchingFPRAS.OnlineInitial
open TreeTyped TreeFraction OnlineSchedule Polynomial NumericBounds ExecutableSource ExecutableScaling
lemma work_data : WorkBound dataRealizer configSize := by
have lift {A : Type} [Codec A] {f : Config→A} {F : Realizer f} (hF : PolynomialTime F) :
WorkBound F configSize := work_of_polynomial hF X (fun x=>by simpa using weight_configSize x)
have wb : ∀ x : Config,weight x≤X.eval (configSize x) := fun x=>by simpa using weight_configSize x
have _workK : WorkBound (first.snd : Realizer (fun x : Config=>x.1.2)) configSize := lift (by unfold Realizer.snd; tree_cost)
have hk : Bound (fun x : Config=>x.1.2) configSize := of_le (fun x=>(config_fields x).2.1)
have hN : Bound (fun x : Config=>x.2.1) configSize := of_le (fun x=>(config_fields x).2.2.1)
have hD : Bound (fun x : Config=>x.2.2) configSize := of_le (fun x=>(config_fields x).2.2.2.1)
have hT : Bound (fun x : Config=>ladderSchedule (x.2,x.1.2)) configSize := of_le (fun x=>(config_fields x).2.2.2.2.1)
have hq : Bound (fun x : Config=>Transport.ladderQ x.2.1 x.2.2) configSize := of_le (fun x=>(config_fields x).2.2.2.2.2)
unfold dataRealizer Realizer.fst Realizer.snd
apply work_congr
apply work_pair _ _ X wb
· apply work_pair _ _ X wb
· exact lift (by tree_cost)
· apply work_pair _ _ X wb
· apply work_pair _ (lift (by tree_cost)) X wb
apply work_pair _ _ X wb
· apply work_comp_bound (lift (by apply polynomial_addNat (polynomial_mulNat (polynomial_constant _) (by tree_cost)) (polynomial_constant _))) fuel_numeric_work
exact add (mul (const 2 _) hk) (const 2 _)
· apply work_comp_bound (lift (by apply polynomial_mulNat (polynomial_constant _) (by tree_cost))) fuel_numeric_work
exact mul (const 2 _) hk
· apply work_pair _ _ X wb
· apply work_comp_bound _ fuel_numeric_work hT
apply work_comp_bound (lift (by tree_cost)) work_ladderSchedule
exact add (add (add hN hD) hk) (NumericBounds.weight (lift (by tree_cost)))
· apply work_comp_bound (lift (polynomial_composition polynomial_second polynomial_q)) fuel_numeric_work hq
· exact lift (polynomial_composition (by tree_cost) polynomial_scaleVector)
lemma work_trialInput : WorkBound trialInputRealizer configSize := by
have lift {A : Type} [Codec A] {f : Config→A} {F : Realizer f} (hF : PolynomialTime F) :
WorkBound F configSize := work_of_polynomial hF X (fun x=>by simpa using weight_configSize x)
have wb : ∀ x : Config,weight x≤X.eval (configSize x) := fun x=>by simpa using weight_configSize x
unfold trialInputRealizer Realizer.snd
apply work_pair _ _ X wb
· exact work_comp_bound (lift (by tree_cost)) fuel_numeric_work (of_le (fun x=>(config_fields x).2.1))
· apply work_pair _ _ X wb
· exact work_pair work_data (lift polynomial_count) X wb
· apply work_pair _ (lift (polynomial_constant _)) X wb
apply work_comp_bound (lift (by unfold nRealizer Realizer.fst;tree_cost)) pairingNumber_numeric_work
exact of_le (fun x=>(config_fields x).1)
lemma work_runInput : WorkBound runInputRealizer configSize := by
unfold runInputRealizer Realizer.fst Realizer.snd
apply work_congr
apply work_pair _ work_trialInput X (fun x=>by simpa using weight_configSize x)
apply work_of_polynomial (polynomial_composition (by tree_cost) polynomial_repetitions) X
exact fun x=>by simpa using weight_configSize x
lemma work_initializeInput : WorkBound initializeInputRealizer seedSize :=
work_comp_bound work_config work_runInput bound_configSize
end MatchingFPRAS.OnlineInitial
end OAI
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